Rank of Unit Group of Burnside Ring: Calculation by Table of Marks

After studying group action, orbit & stabilizer theorem, and isomorphism class during the second week, I began to have a clearer picture of how to approach calculating the rank of the unit groups of Burnside rings. My goal for the third week is to find the methodology of calculating the rank of the unit group of a Burnside ring by hand, which would later shed some light on my later work of automating the process in GAP. During the third week, I read two research paper, which are “On the Unit Groups of Burnside Rings” by Tomboyish Yoshida, and “An Algorithms for the Unit Group of the Burnside Ring of A Finite Group” by Robert Boltje and Gotz Sniffer. There were very complicated parts of the paper that I could not fully comprehend, so I mainly focused on understanding the parts that are more crucial to my project. I also proved below propositions.

propositions week 3

At the end of week 3, I eventually found the way of to calculate the rank of the unit group of Burnside rings of smaller and simpler groups, such as C2, C3, and D4 by using the idea of table of marks and ghost ring. I practiced solving for C2, C3, and D4 by hand. (method shown in graph below)

Practice solving for rank of U of B(G) Copy

 

In conclusion, I achieved the goal of finding the way of calculating the rank of the unit group of Burnside ring during Week 3. The next step of the project to convert the work on paper into a GAP function and test the efficiency of the function.